# Is there any connection between frequency response of a system and spectrum of a signal?

For instance, an AR process can be generated through a white noise $w[n]$ passing through an all-poles system (IIR), $$y[n] = \sum_{k = 1}^{p}a_ky[n-k] + w[n]$$ and we can analyze the IIR system, instead of obtaining the spectrum of the signal, to get some properties of the signal and therefore apply it to some applications, e.g., prediction.

My question is whether all the signals, stationary or non-stationary, can be treated as it is generated via some system driven by white noise, and we can analyze the system, which hopefully can completely characterize the signals.

• Can I please ask you to clarify your question? Systems generate signals. Features of the signals can be tied back to what the system was doing to produce this signal. Therefore, yes, there is a connection between the frequency response of a system and the spectrum of a signal that was acquired in some way from that system. This, I think, is a trivial answer. I am not so sure that this is what you are after though. – A_A Nov 22 '17 at 8:41
• look at Wold decomposition theorem. – Stanley Pawlukiewicz Nov 23 '17 at 3:00
• Thank you @ Stanley Pawlukiewicz. That was very enlightened! – ZHUANG Nov 23 '17 at 3:03
• too bad I couldn’t give a proper answer – Stanley Pawlukiewicz Nov 23 '17 at 3:15
• @StanleyPawlukiewicz You can still provide a slightly more extensive response to the question and Bruce zhuang can accept it so that the question is closed gracefully and stops being recycled on the board. – A_A Nov 24 '17 at 9:43

Consider an LTI system with impulse response $h[n]$. When the input $x[n]$ to the system is assumed to be an instance of a WSS random process, then the ouptut $y[n]$ can also be shown to be belonging to a WSS random process, with the following relation between their ACFs and power spectrums. $$y[n] = h[n] \star x[n] \implies \phi_{yy}[m] = h[m] \star h[-m]^* \star \phi_{xx}[m]$$ where $\phi_{xx}[m]$ and $\phi_{yy}[m]$ are the ACF sequences of the input and output sequences respectively. Then by the theorem which states the relation between the ACF and PSD of the WSS random processes. $$\Phi_{xx}(\omega) = \mathcal{FT} \{ \phi_{xx}[m] \} = \sum_{m=-\infty}^{\infty} \phi_{xx}[m] e^{-j \omega m}$$ one can see that: $$\phi_{yy}[m] = h[m] \star h[-m]^* \star \phi_{xx}[m] \implies \Phi_{yy}(\omega) = |H(e^{j \omega})|^2 \Phi_{xx}(\omega)$$ Which sates the relation between the WSS input and ouput PSDs and frequency resplonse $H(e^{j \omega})$ of the LTI system.

As mentioned in my comment, look at the Wold Decomposition Theorem :

https://en.wikipedia.org/wiki/Wold%27s_theorem

but actually showing how it relates to your question,

In simpler terms, your AR system can be represented by a MA system (possibly infinite order) and as the second reference shows, an ARMA system. The process has more than one representation so there isn't a one to one correspondence between the time series and some system that generated it.

So the answer is yes, there is a correspondence and often the system itself, is of interest, but the time series and system are not uniquely equivalent.

another way of thinking about it is that $H(\omega)$ has a phase and magnitude response that gives $|H(\omega)|^2$ but there are other possible phase responses that results in the same magnitude squared.